1. PreCalculus Section 2.7 Apply general theorems about polynomials Fundamental Theorem of Algebra In the complex number system, a polynomial of degree n has exactly n roots.(cubic has three roots, quartic has four roots, etc..) How many roots does P(x) = 3x 5 – 6x 2 + 9 have? Answer: five

2. Complex Conjugates Theorem If a polynomial with real coefficients has the complex root a + bi, then it also has a – bi as a root. If 7 – 6i is a root of a polynomial with real coefficients, what is another of its roots? Answer: 7 + 6i Irrational Conjugates Theorem If a polynomial with rational coefficients has the irrational root a + √b then it also has the root a - √b If 3 + √2 is a root of a polynomial with rational coefficients, then 3 - √2 is also a root.

3. Odd degree Theorem If a polynomial has real coefficients and an odd degree, then it has at least one real root.

4. For the equation ax n + bx n-1 + cx n-2 + … dx + e The sum of the roots is given by: -b/a Find the sum of the roots of 3x 4 + 2x 3 - 6x 2 + 7x + 9 Answer: -2/3 The product of the roots is given by: e/a if n is even -e/a if n is odd Find the product of the roots of 3x 4 + 2x 3 - 6x 2 + 7x + 9 Answer: 9/3 = 3

5. Quadratic Equation Theorem A quadratic equation can be written as: x 2 – (sum of the roots)x +(product of the roots) = 0 Find the equation of the quadratic equation with roots of 3 + 2i and 3 – 2i. Find the equation of the cubic equation with roots of 5 - √3, 5 + √3, and 6.

6. assignment Page 89 Problems 1-10,13,14,18,19,20,24